The long and short is that the Bill Clinton and his family are not descended from the Clintons who were prominent in New York politics in the late 18th and early 19th century.

Part of this is because Bill Clinton wasn't born a "Clinton" at all - but as William Jefferson Blythe. His mother remarried to Roger Clinton Sr. when Bill was young, and he formally changed his surname in his teens. Roger Clinton's ancestors had lived in the Deep South for a couple centuries and were not directly descended from the New York Clintons (who were established in New York by Colonel Charles Clinton in the late 17th century).

The name (both a first name and surname) "Clinton" is relatively common in England and comes from a few localities, so it is not related to a common ancestry. This led to another surname coincidence worth mentioning - namely, that General Henry Clinton was the commander in charge of the successful British capture of New York City in 1776, and while he had spend previous time living in New York, he was not related to the New York Clintons - George Clinton becoming the patriot Governor of New York in 1777.